A note on the long-time behavior of dissipative solutions to the Euler system. (English) Zbl 07451387

Summary: We show that the Reynolds defect measure for a dissipative weak solution of the compressible Euler system vanishes for large time. This may be seen as a piece of evidence that the dissipative solutions are asymptotically close to weak solutions in the turbulent regime, whence suitable for describing compressible fluid flows in the long run.


35Q31 Euler equations
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[1] D. Breit, E. Feireisl, and M. Hofmanová. Dissipative solutions and semiflow selection for the complete Euler system. Comm. Math. Phys., 376(2):1471-1497, 2020. · Zbl 1441.35188
[2] D. Breit, E. Feireisl, and M. Hofmanová. Solution semiflow to the isentropic Euler system. Arch. Ration. Mech. Anal., 235(1):167-194, 2020. · Zbl 1441.35164
[3] Chiodaroli, E., A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Equ., 11, 3, 493-519 (2014) · Zbl 1304.35515
[4] Chiodaroli, E.; De Lellis, C.; Kreml, O., Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math., 68, 7, 1157-1190 (2015) · Zbl 1323.35137
[5] Chiodaroli, E.; Kreml, O., On the energy dissipation rate of solutions to the compressible isentropic Euler system, Arch. Ration. Mech. Anal., 214, 3, 1019-1049 (2014) · Zbl 1304.35516
[6] E. Chiodaroli, O. Kreml, V. Mácha, and S. Schwarzacher. Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data. Trans. Amer. Math. Soc., 374(4): 2269-2295, 2021. · Zbl 1462.35263
[7] Dafermos, CM, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations, 14, 202-212 (1973) · Zbl 0262.35038
[8] Dafermos, CM, Maximal dissipation in equations of evolution, J. Differential Equations, 252, 1, 567-587 (2012) · Zbl 1242.35167
[9] Dafermos, CM, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70, 167-179 (1979) · Zbl 0448.73004
[10] E. Feireisl and M. Hofmanová. On convergence of approximate solutions to the compressible Euler system. Ann. PDE, 6(2):24, 2020. (Paper No. 11) · Zbl 1448.35375
[11] E. Feireisl, M. Lukáčová-Medvidová, and H. Mizerová. Convergence of finite volume schemes for the Euler equations via dissipative measure-valued solutions. Found. Comput. Math., 20(4):923-966, 2020. · Zbl 1447.65050
[12] E. Feireisl, M. Lukáčová-Medvidová, H. Mizerová, B. She, and Wang. Computing oscillatory solutions to the Euler system via K-convergence. 2019. arxiv preprint No.arXiv:1910.03161.
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